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Algebra Difficulty 5.0 AIME Prove it North Macedonia

Mile imagined a number and said: "If I multiply the number by two and add this product to the half of the number I will get a number that is for 26 bigger than the third of the imagined number." What number did Mile imagine?

Solution

Let xx be the number that Mile imagined. Then we solve the equation 2x+12x=13x+262x + \frac{1}{2}x = \frac{1}{3}x + 26, from where we get 12x+3x=2x+15612x + 3x = 2x + 156 or 13x=15613x = 156, x=12x = 12. Mile imagined the number 12.

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